Yes, the near-identical sentences (only drawing a distinction between the processes where one exists) would indicate that. The "heard by the bird" and "reproduced by the bird" steps were also the same. But this is necessary context to make clear the digital data ("bit-stream") that is being modulated into the signal.
It is far from "exactly the same". The similarity is only in that both go through the same analogue channel. The entire point is that the modulated signal can be reconstructed exactly, while the spectrogram cannot.
The article title says they converted a PNG and the bird was able to "recall the file", and yet it produced an indisputably different file. That it looks vaguely the same to the cursory human observer does not make it the same file.